Mathworks COMMUNICATIONS TOOLBOX 4 Reference

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Communications To

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Communications Toolbox™ Reference

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Revision History

April 1996 First printing Version 1.0 May 1997 Second printing Revised for Version 1.1 (MATLAB 5.0) September 2000 Third printing Revised for Version 2.0 (Release 12) May 2001 Online only Revised for Version 2.0.1 (Release 12.1) July 2002 Fourth printing Revised for Version 2.1 (Release 13) June 2004 Fifth printing Revised for Version 3.0 (Release 14) October 2004 Online only Revised for Version 3.0.1 (Release 14SP1) March 2005 Online only Revised for Version 3.1 (Release 14SP2) September 2005 Online only Revised for Version 3.2 (Release 14SP3) October 2005 Reprint Version 3.0 (Notice updated) March 2006 Online only Revised for Version 3.3 (Release 2006a) September 2006 Sixth printing Revised for Version 3.4 (Release 2006b) March 2007 Online only Revised for Version 3.5 (Release 2007a) September 2007 Online only Revised for Version 4.0 (Release 2007b) March 2008 Online only Revised for Version 4.1 (Release 2008a) October 2008 Online only Revised for Version 4.2 (Release 2008b) March 2009 Online only Revised for Version 4.3 (Release 2009a) September 2009 Online only Revised for Version 4.4 (Release 2009b) March 2010 Online only Revised for Version 4.5 (Release 2010a)

Function Reference

Signal Sources .................................... 1-2

Contents

Performance Evaluation

Source Coding

Error-Control Coding

Interleaving/Deinterleaving

Analog Modulation/Demodulation

Digital Modulation/Demodulation

Pulse Shaping

Filters

Lower-Level Filters

Channels

Equalizers

............................................ 1-9

..................................... 1-3

..................................... 1-9

.......................................... 1-9

........................................ 1-11

........................... 1-2

.............................. 1-4

........................ 1-6

................................ 1-9

................... 1-7

Galois Field

MATLAB F unctions and Operators

Galois Fields of Odd Characteristic

Utilities

MATLAB U tilities

....................................... 1-11

........................................... 1-14

................................. 1-16

................... 1-12

................. 1-13

GUI .............................................. 1-16

Functions — Alphabetical List

Index

vi Contents

Function Reference

Signal Sources (p. 1-2) Sources of random signals

Performance Evaluation (p. 1-2) Analyzing and visualizing

performance of a communication system

Source Coding (p. 1-3) Quantization, companders, and

other kinds of source coding

Error-Control Coding (p. 1-4)

Interleaving/Deinterleaving (p. 1-6)

Analog Modulation/Demodulation (p. 1-7)

Digital Modulation/Demodulation (p. 1-7)

Pulse Shaping (p. 1-9) Oversampling and shaping a signal

Filters (p. 1-9) Raised cosine and Hilbert filters

Channels (p. 1-9) Channel models for real, complex,

Equalizers (p. 1-11) Adaptive and MLSE equalizers

Galois Field (p. 1-11) Manipulating elements of finite

Galois Fields of Odd Characteristic (p. 1-13)

Utilities (p. 1-14) Miscellaneous relevant functions

GUI (p. 1-16)

Block and convolutional coding

Block and convolutional interleaving

Passband amplitude, frequency, and phase modulation

Baseband digital modulation

and binary signals

fields of even order

Manipulating elements of finite fields of odd order

Bit error rate analysis tool

1 Function Reference

Signal Sources

Performan

commsrc.pattern

randerr

randint

randsrc

wgn

ce Evaluation

berawgn

bercoding

berconfint

berfading

berfit

sync

ber

terr

ommmeasure.ACPR

commmeasure.EVM

commmeasure.MER

Construct pattern generator object

Generate bit error patterns

Generate matrix of uniformly distributed random integers

Generate random matrix using prescribed alphabet

Generate white Gaussian noise

Bit error rate (BER) for uncoded AWGN channels

Bit error rate (BER) for coded AWGN channels

Bit error rate (BER) and confidence interval of Monte Carlo simulation

Bit error rate (BER) for Rayleigh and Rician fa ding channels

urve to nonsmooth empirical bit

Fit c

r rate (BER) data

erro

error rate (BER) for imperfect

Bit

chronization

syn

mpute number of bit errors and

t error rate (BER)

eate adjacent channel power

easurement object

reate EVM measurement object

Create MER measurement object

1-2

Source Coding

commscope

commscope.eyediagram

commscope.ScatterPlot

commtest.ErrorRate

distspec

eyediagram

EyeScope

noisebw

scatterplot

semianalytic

symerr

Package of communications scope classes

Eye diagram analysis

Create Scatter Plot scope

Create error rate test console

Compute distance spectrum of convolutional code

Generate eye diagram

Launch eye diagram scope for eye diagram object H

Equivalent noise bandwidth of filter

Generate scatter plot

Calculate bit error rate (BER) using semianalytic technique

Compute number of symbol errors and symbol error rate

Source Coding

arithdeco

arithenco

compand

dpcmdeco

dpcmenco

Decode binary code using arithmetic decoding

Encode sequence of symbols using arithmetic coding

Source code mu-law or A-law compressor or expander

Decode using differential pulse code modulation

Encode using differential pulse code modulation

1-3

1 Function Reference

dpcmopt

huffmandeco

huffmandict

huffmanenco

lloyds

quantiz

Error-Control Coding

bchdec

bchenc

bchgenpoly

bchnumerr

convenc

cyclgen

cyclpoly

decode

dvbs2ldpc

encode

fec.bchdec

Optimize differential pulse code modulation parameters

Huffman decoder

Generate Huffman code dictionary for source with known probability model

Huffman encoder

Optimize quantization parameters using Lloyd algorithm

Produce quantization index and quantized output value

BCH decoder

BCH encoder

Generator polynomial of BCH code

Number of correctable errors for BCH code

Convolutionally encode binary data

Produce parity-check and generator matrices for cyclic code

Produce generator po lyno mials for cyclic code

Block decoder

Low-density parity-check codes from DVB-S.2 standard

Block encoder

Construct B CH decoder object

1-4

Error-Control Coding

fec.bchenc

fec.ldpcdec

fec.ldpcenc

fec.rsdec

fec.rsenc

gen2par

gfweight

hammgen

rsdec

rsdecof

rsenc

rsencof

rsgenpoly

syndtable

vitdec

Construct BCH encoder object

Construct LD PC decoder object

Construct LDPC encoder object

Construct Reed-Solomon decoder object

Construct Reed-Solomon encoder object

Convert between parity-check and generator matrices

Calculate minimum distance of linear block code

Produce parity-check and generator matrices for Hamming code

Reed-Solomon decoder

Decode ASCII file encoded using Reed-Solomon code

Reed-Solomon encoder

Encode ASCII file using Reed-Solomon code

Generator polynomial of Reed-Solomon code

Produce syndrome decoding table

Convolutionally decode binary data using Viterbi al gorithm

1-5

1 Function Reference

Interleaving/Deinterleaving

algdeintrlv

algintrlv

convdeintrlv

convintrlv

deintrlv

heldeintrlv

helintrlv

helsca

helsc

intrlv

matdeintrlv

matintrlv

muxdeintrlv

muxintrlv

randdeintrlv

randintrlv

ndeintrlv

anintrlv

Restore ordering of symbols using algebraically derived permutation table

Reorder symbols using algebraically derived permutation table

Restore ordering of symbols using shift registers

Permute symbols using shift registers

Restore ordering of symbols

Restore ordering of symbols permuted using

Permute

Restor helica

symbols using helical array

e ordering of symbols in

l pattern

helintrlv

Reorder symbols in helical pattern

Reorder sequence of symbols

Restore ordering of symbols by filling matrix by columns and emptying it by rows

Reorder symbols by filling matrix by rowsandemptyingitbycolumns

Restore ordering of symbols using specified shift registers

Permute symbols using shift registers with specified delays

Restore ordering of symbols using random permutation

Reorder symbols using random permutation

1-6

Analog Modulation/Demodulation

amdemod

ammod

fmdemod

fmmod

pmdemod

pmmod

ssbdemod

ssbmod

Digital Modulation/Demodulation

emod

dpskd

mod

dpsk

demod

fsk

mod

fsk

nqamdemod

enqammod

modem

modem.dpskdemod

modem.dpskmod

Amplitude demodulation

Amplitude m odulation

Frequency demodulation

Frequency mo

dulation

Phase demodulation

Phase modulation

Single sideband am plitude demodulation

Single sideband am plitude modulation

rential phase shift keying

Diffe demod

Diff modu

Fre

Ge de

G m

ulation

erential phase shift keying

lation

quency shift keying demodulation

quency shift keying modulation

neral quadrature amplitude

modulation

eneral quadrature amplitude

odulation

Package of modem classes

Construct DPSK demodulator object

Construct DPSK modulator object

1-7

1 Function Reference

modem.genqamdemod

modem.genqammod

modem.mskdemod

modem.mskmod

modem.oqpskdemod

modem.oqpskmod

modem.pamdemod

modem.pammod

modem.pskdemod

modem.pskmod

modem.qamdemod

modem.qammod

modnorm

mskdemod

mskmod

oqpskdemod

oqpskmod

pamdemod

pammod

pskdemod

pskmod

qamdemod

qammod

Construct General QAM demodulator object

Construct General QAM modulator object

Construct MSK demodulator object

Construct MSK modulator object

Construct OQPSK demodulator object

Construct OQPSK modulator object

Construct PAM demodulator object

Construct PAM modulator object

Construct PSK demodulator object

Construct PSK modulator object

Construct QAM demodulator object

Construct QAM modulator object

Scaling factor for normalizing modulation output

Minimum shift keying demodulation

Minimum shift keying modulation

Offset quadrature phase shift keying demodulation

Offset quadrature phase shift keying modulation

Pulse amplitude demodulation

Pulse amplitude modulation

Phase shift keying demodulation

Phase shift keying modulation

Quadrature amplitude demodulation

Quadrature amplitude modulation

1-8

Pulse Shaping

Filters

intdump

rcosflt

rectpulse

hank2sys

hilbiir

rcosine

Lower-L

rcosfir

rcosiir

evel Filters

Integrate and dump

Filter input si cosine filter

Rectangular pulse shaping

Convert Hankel matrix to linear system model

Design Hilbert transform IIR filter

Design raised cosine filter

Design raised cosine finite impulse response (FIR) filter

Design raised cosine infinite impulse response (IIR) filter

gnal using raised

Channels

awg

bsc

doppler

doppler.ajakes

white Gaussian noise to signal

Add

Model binary symmetric channel

Package of Doppler classes

Construct asymmetrical Doppler spectrum object

1-9

1 Function Reference

doppler.bell

doppler.bigaussian

doppler.flat

doppler.gaussian

doppler.jakes

doppler.rjakes

doppler.rounded

filter (channel)

legacychannelsim

mimochan

plot (channel)

rayleighchan

reset (channel)

ricianchan

stdchan

Construct bell-shaped Doppler spectrum object

Construct bi-Gaussian Doppler spectrum object

Construct flat Doppler spectrum object

Construct Gaussian Doppler spectrum object

Construct Jakes Doppler spectrum object

Construct restricted Jakes Doppler spectrum object

Construct rounded Doppler spectrum object

Filter sig nal w i th channel object

Toggles random number generation mode for channel objects

Create MIMO fading channel object

Plot channel characteristics with channel v isualization tool

Construct Rayleigh fading channel object

Reset channel object

Construct Rician fading channel object

Construct channel object from set of standardized channel models

1-10

Equalizers

cma

dfe

equalize

lineareq

lms

mlseeq

normlms

reset (

rls

signlms

varlms

equalizer)

Construct constant modulus algorithm (CMA) object

Construct decision-feedback equalizer object

Equalize signal using equalizer object

Construct linear equalizer object

Construct least mean square (LMS) adaptive algorithm object

Equalize using Vit

Constru square ( object

Reset equalizer object

Construct recursive least squares (RLS) adaptive algorithm object

Construct signed least m ean square (LMS) a daptive algorithm object

Construct variable-step-size least mean square (LMS) adaptive algorithm object

linearly modulated signal

erbi algorithm

ct normalized least mean LMS) adaptive algorithm

alois Field

convmtx

cosets

Convolution matrix of Galois field vector

Produce cyclo tomic cosets for Galois field

1-11

1 Function Reference

dftmtx

Discrete Fourier transform matrix in Galois field

fft

filter (gf)

gftable

Discrete Fourier transform

1-D digital filter over Galois field

Create Galois field array

Generate file to accelerate Galois field computations

ifft

isprimitive

Inverse discrete Fourier transform

True for primitive polynomial for Galois field

log

minpol

Logarithm in Galois field

Find minimal polynomial of Galois field element

mldivide

primpoly

Matrix left division \ of Galois arrays

Find primitive polynomials for Galois field

MATLAB Functions and Operators

1-12

*/\

.* ./ .\

'.'

==, ~=

all

any

conv

Addition and subtraction of Galois arrays

Matrix multiplication and d iv isi o n of Galois arrays

Elementwise m ultiplicat ion and division of Galois arrays

Matrix exponentiation of Galois array

Elementwise e xponentiation of Galois array

Transpose of Galois array

Relational o perators for Galois arrays

True if all elements o f a Galois vector are n onzero

True if any element of a Galois vector is nonzero

Convolution of Galois vectors

Galois Fields of Odd Characteristic

deconv Deconvolution and polynomial division

det

diag

inv

isempty

length

polyval

rank

reshape

roots

size

tril

triu

Determinant of square G alois matrix

Diagonal Galois matrices and diagonals of a Galois matrix

Inverse of Galois matrix

True for empty Galois arrays

Length of Galois vector

Lower-upper triangular factorization of Galois array

Evaluate polynomial in Galois field

Rank of a Galois array

Reshape Galois array

Find poly nomial roots across a Galois field

Size of Galois array

Extract low er triangular part of Galois array

Extract upper triangular part of Galois array

Galois Fields of Odd Characteristic

gfadd

gfconv

gfcosets

gfdeconv

gfdiv

gffilter

Add polyn omials over Galois field

Multiply polynomials over Galois field

Produce cyclo tomic cosets for Galois field

Divide polynomials over Galois field

Divide eleme nts of Galois field

Filter data using polynomials over prime Galois field

1-13

1 Function Reference

gflineq

gfminpol

gfmul

gfpretty

gfprimck

gfprimdf

gfprimfd

gfrank

gfrepcov

gfroots

gfsub

gftrunc

gftuple

Find particular solution of Ax = b over prime Galois field

Find minimal polynomial of Galois field element

Multiply elements of G a lois field

Polynomial in traditional format

Check whether polynomial over Galois field is primitive

Provide defau lt primitive polynomials for Galois field

Find primitive polynomials for Galois field

Compute rank of m atrix over Galois field

Convert one binary polynomial representation to another

Find roots of polynomial over prime Galois field

Subtract polynomials over Galois field

Minimize length of polynomial representation

Simplify or convert Galois field element formatting

Utilities

1-14

alignsignals

bi2de

Align two signals b y delaying earliest signal

Convert binary vectors to decimal numbers

Utilities

bin2gray

commsrc.pn

de2bi

finddelay

gray2bin

iscatastrophic

istrellis

marcumq

mask2shift

oct2dec

poly2trellis

qfunc

qfuncinv

seqgen

seqgen.pn

shift2mask

vec2mat

Convert positive integers into corresponding Gray-encoded integers

Create PN sequence g enerator package

Convert decimal numbers to binary vectors

Estimate delay(s) between signals

Convert Gray-encoded positive integers to corresponding Gray-decoded integers

True for trellis corresponding to catastrophic convolutional code

True for valid trellis structure

Generalized M arcum Q function

Convert mask vector to shift for shift register configuration

Convert octal to decimal numbers

Convert convolutional code polynomials to trellis description

Qfunction

Inverse Q function

Sequence generator package

Construct default PN sequence generator object

Convert shift to mask vector for shift register configuration

Convert vector into matrix

1-15

1 Function Reference

GUI

MATLAB Utilitie

erf Error function

erfc Complementary error function

bertool

Open bit error rate analysis GUI (BERTool)

1-16

Functions — Alphabetical List

algdeintrlv

Purpose Restore ordering of symbols using algebraically derived permutation

table

Syntax deintrlvd = algdeintrlv(data,num,'takeshita-costello',k,h)

deintrlvd = algdeintrlv(data,num,

Description deintrlvd = algdeintrlv(data,num,'takeshita-costello',k,h)

restores the original ordering of the elements in data using a permutation table that is algebraically derived using the Takeshita-Costello method.

data is a vector, or the number of rows of data if data is a matrix with

multiple columns. In the Takeshita-C ostello method, power of 2. The multiplicative factor, than

num,andthecyclicshift,h, must be a nonnegative integer less than

num.Ifdata is a matrix with multiple rows and columns, the function

processes the columns independently.

deintrlvd = algdeintrlv(data,num,'welch-costas',alph) uses the

Welch-Costas method. In the Welch-Costas method, prime number.

alph is an integer between 1 an d num that represents a

primitive element of the finite field GF(

num is the number of elements in data if

'welch-costas',alph)

num must be a

k,mustbeanoddintegerless

num+1 must be a

num+1).

To use this function as an inverse of the same inputs in both functions, except for the

algintrlv function, use the

data input. In that case,

the two functions are inverses in the sense that applying followed by algdeintrlv leaves data unchanged.

Examples The code below uses the Takeshita-Costello method of algintrlv and

algdeintrlv.

num = 16; % Power of 2 ncols = 3; % Number of columns of data to interleave data = rand(num,ncols); % Random data to interleave k=3; h=4; intdata = algintrlv(data,num,'takeshita-costello',k,h); deintdata = algdeintrlv(intdata,num,'takeshita-costello',k,h);

2-2

algintrlv

algdeintrlv

See Also finddelay

X=[123]; Y=[0000123]; [Xa Ya D] = alignsignals(X, Y, [], 'truncate')

2-10

amdemod

Purpose Amplitude demodulation

Syntax z = amdemod(y,Fc,Fs)

z = amdemod(y,Fc,Fs,ini_phase) z = amdemod(y,Fc,Fs,ini_phase,carramp) z = amdemod(y,Fc,Fs,ini_phase,carramp,num,den)

Description z = amdemod(y,Fc,Fs) demodulates the amplitude modulated signal y

from a carrier signal with frequency Fc (Hz). The carrier signal and y have sample frequency Fs (Hz). The m odulated signal y has zero initial phase and zero carrier amplitude, so it represents suppressed carrier modulation. The demodulation processusesthelowpassfilterspecified by

[num,den] = butter(5,Fc*2/Fs).

Note The Fc and Fs arguments must satisfy Fs >2(Fc + BW), where BW is the bandwidth of the originalsignalthatwasmodulated.

z = amdemod(y,Fc,Fs,ini_phase) specifies the initial phase of the

modulated signal in radians.

z = amdemod(y,Fc,Fs,ini_phase,carramp) demodulates a signal that

was created via transmitted carrier modulation instead of suppressed carrier modulation. signal.

z = amdemod(y,Fc,Fs,ini_phase,carramp,num,den) specifies

the numerator and denominator of the lowpass filter used in the demodulation.

carramp is the carrier am plitude of the modulated

Examples The code below illustrates the use of a nondefault filter.

t = .01; Fc = 10000; Fs = 80000; t = [0:1/Fs:0.01]'; s = sin(2*pi*300*t)+2*sin(2*pi*600*t); % Original signal

2-11

amdemod

[num,den] = butter(10,Fc*2/Fs); % Lowpass filter

y1 = ammod(s,Fc,Fs); % Modulate. s1 = amdemod(y1,Fc,Fs,0,0,num,den); % Demodulate.

See Also ammod, ssbdemod, fmdemod, pmdemod,“Modulation”

2-12

ammod

Purpose Amplitude modulation

Syntax y = ammod(x,Fc,Fs)

y = ammod(x,Fc,Fs,ini_phase) y = ammod(x,Fc,Fs,ini_phase,carramp)

Description y = ammod(x,Fc,Fs) uses the message signal x to modulate a carrier

signal with frequency signal and has zero initial phase and zero carrier amplitude, so the result is suppressed-carrier modulation.

Note The x, Fc,andFs input arguments must satisfy Fs >2(Fc + BW), where

y = ammod(x,Fc,Fs,ini_phase) specifies the initial phase in the

modulated signal

x have sample frequency Fs (Hz). The modulated signal

BW is the bandwidth of the modulating signal x.

Fc (Hz) using amplitude modulation. The carrier

y in radians.

y = ammod(x,Fc,Fs,ini_phase,carramp) performs

transmitted-carrier modulation instead of suppressed-carrier modulation. The carrier amplitude is

carramp.

Examples The example below co m p are s double-sideband and single-sideband

amplitude modulation.

% Sample the signal 100 times per second, for 2 seconds. Fs = 100; t = [0:2*Fs+1]'/Fs; Fc = 10; % Carrier frequency x = sin(2*pi*t); % Sinusoidal signal

% Modulate x using single- and double-sideband AM. ydouble = ammod(x,Fc,Fs); ysingle = ssbmod(x,Fc,Fs);

2-13

ammod

% Compute spectra of both modulated signals. zdouble = fft(ydouble); zdouble = abs(zdouble(1:length(zdouble)/2+1)); frqdouble = [0:length(zdouble)-1]*Fs/length(zdouble)/2; zsingle = fft(ysingle); zsingle = abs(zsingle(1:length(zsingle)/2+1)); frqsingle = [0:length(zsingle)-1]*Fs/length(zsingle)/2;

% Plot spectra of both modulated signals. figure; subplot(2,1,1); plot(frqdouble,zdouble); title('Spectrum of double-sideband signal'); subplot(2,1,2); plot(frqsingle,zsingle); title('Spectrum of single-sideband signal');

See Also amdemod, ssbmod, fmmod, pmmod,“Modulation”

2-14

arithdeco

Purpose Decode binary code using arithmetic decoding

Syntax dseq = arithdeco(code,counts,len)

Description dseq = arithdeco(code,counts,len) decodes the binary arithmetic

code in the vector symbols. The vector counts represents the source’s statistics by listing the number of times each symbol of the source’s alphabet occurs in a test data set. This function assumes that the data in by the

arithenco function.

Examples This example is similar to the example on the arithenco reference

page,exceptthatituses

counts = [99 1]; % A one occurs 99% of the time. len = 1000; seq = randsrc(1,len,[1 2; .99 .01]); % Random sequence code = arithenco(seq,counts); dseq = arithdeco(code,counts,length(seq)); % Decode. isequal(seq,dseq) % Check that dseq matches the original seq.

code to recover the corresponding sequence of len

code was produced

arithdeco to recover the original sequence.

The output is

ans =

Algorithm This function uses the algorithm described in [1].

See Also arithenco,“ArithmeticCoding”

References [1] Sayood, Khalid, Introduction to Data Compression, San Francisco,

Morgan Kaufmann, 2000.

2-15

arithenco

Purpose Encode sequence of symbols using arithmetic coding

Syntax code = arithenco(seq,counts)

Description code = arithenco(seq,counts) generates the binary arithmetic

code corresponding to the sequence of symbols specified in the vector

seq.Thevectorcounts represents the source’s statistics by listing t he

number of times each symbol of the source’s alphabet occurs in a test data set.

Examples This example illustrates the compression that arithmetic coding can

accomplish in some situations. A source has a two-symbol alphabet and produces a test data set in which 99% of the sy mbols are 1s. Encoding 1000 symbols from this source produces a code vector having many fewer than 1000 elements. The actual number of elements in varies, depending on the particular random sequence contained in seq.

counts = [99 1]; % A one occurs 99% of the time. len = 1000; seq = randsrc(1,len,[1 2; .99 .01]); % Random sequence code = arithenco(seq,counts); s = size(code) % length of code is only 8.3% of length of seq.

code

The output is

183

Algorithm This function uses the algorithm described in [1].

See Also arithdeco,“ArithmeticCoding”

References [1] Sayood, Khalid, Introduction to Data Compression, San Francisco,

Morgan Kaufmann, 2000.

2-16

Purpose Add white Gaussian noise to signal

Syntax y = awgn(x,snr)

y = awgn(x,snr,sigpower) y = awgn(x,snr,' y = awgn(x,snr,sigpower,s) y = awgn(x,snr,' y = awgn(...,powertype)

measured')

measured',state)

Description y = awgn(x,snr) adds white Gaussian noise to the vector signal x.

The scalar

x is complex, awgn adds complex noise. This syntax assumes that the

power of

y = awgn(x,snr,sigpower) is the same as the syntax above, except

that

y = awgn(x,snr,'measured') isthesameasy = awgn(x,snr),except

that

y = awgn(x,snr,sigpower,s) uses s, which is a random stream

handle, to generate random noise samples with randn. If s is an integer, then resets the state of randn to s. The latter usage is obsolete and may be removed in a future release. If you want to generate repeateable noise samples, then provide the handle of a random stream or use reset method on the default random stream.

snr specifies the signal-to-noise ratio per sample, in dB. If

x is 0 dBW.

sigpower is the power of x in dBW .

awgn measures the power of x before adding noise.

awgn

y = awgn(x,snr,'measured',state) is the same as y= awgn(x,snr,'

normal random number generator

measured'),exceptthatawgn first resets the state of

randn to the integer state.

Note This usage is deprecated and may be removed in a future release. Instead of

y = awgn(...,powertype) isthesameastheprevioussyntaxes,

except that the string

sigpower.Choicesforpowertype are 'db' and 'linear'.Ifpowertype

state,uses, as in the previous example.

powertype specifies the units of snr and

2-17

awgn

is 'db',thensnr is measured in dB and sigpower is meas ured in dBW. If

powertype is 'linear', snr is measured as a ratio and sigpower is

measured in watts.

Relationship Among SNR, Es/N0,andEb/N

For the relationships between SNR and other measures of the relative power of the noise, see “Describing the Noise Level of an AWGN Channel”.

Examples The commands below add white Gaussian noise to a sawtooth signal. It

then plots the original and noisy signals.

t = 0:.1:10; x = sawtooth(t); % Create sawtooth signal. y = awgn(x,10,'measured'); % Add white Gaussian noise. plot(t,x,t,y) % Plot both signals. legend('Original signal','Signal with AWGN');

2-18

Several other examples that illustrate the use of awgn are in “Getting Started”. The following demos also use

vitsimdemo,andscattereyedemo.

awgn: basicsimdemo,

See Also wgn, randn, bsc, “AWGN Channel”

awgn

2-19

bchdec

Purpose BCH decoder

Note bchdec will be removed in a future release. Use fec.bchdec

instead.

Syntax decoded = bchdec(code,n,k)

decoded = bchdec(...,paritypos) [decoded,cnumerr] = bchdec(...) [decoded,cnumerr,ccode] = bchdec(...)

Description decoded = bchdec(code,n,k) attempts to decode thereceivedsignal

code using an [n,k] BC H decoder with the narrow-sense generator

polynomial.

n-element row of code represents a corrupted systematic codeword,

where the parity symbols are at the end and the leftmost symbol is the most significant symbol.

code is a Galois array of symbols over GF(2). Each

2-20

In the Galois array decoding the corresponding row in if

bchdec detects more than t errors in a row of code,wheretisthe

number of correctable errors as reported by a decoding failure, merely removing

decoded = bchdec(...,paritypos) specifies whether the parity

symbols in

code were appended or prepended to the message in

the coding operation. The string

'beginning'. The default is 'end'.Ifparitypos is 'beginning',

then a decoding failure causes

decoded, each row represents the attempt at

code.Adecoding failure occurs

bchgenpoly.Inthecaseof

bchdec forms the corresponding row of decoded by

n-k symbols from the end of the row of code.

paritypos can be either 'end' or

bchdec to remove n-k symbols from the

beginning rather than the end of the row.

[decoded,cnumerr] = bchdec(...) returns a column vector cnumerr,

each element of which is the number of corrected errors in the corresponding row of decoding failure in that row in

code.Avalueof-1 in cnumerr indicates a

code.

[decoded,cnumerr,ccode] = bchdec(...) returns ccode,the

corrected version of as

code. If a decoding failure occurs in a certain row of code,the

corresponding row in

code. The Galois array ccode has the same format

ccode contains that row unchanged.

bchdec

Results of Error Correction

BCH decoders correct up to a certain number of errors, specified by the user. If the input contains more errors than the decoder is meant to correct, the decoder will most likely not output the correct codeword.

The chance of a BCH decoder decoding a corrupted input to the correct codeword depends on the number of errors in the input and the number of errors the decoder is meant to correct.

For example, when a single-error-correcting BCH decoder is given input with two errors, it actually decodes it to a different codeword. When a double-error-correcting BCH decoder is given input with three errors, then it only sometimes decodes it to a valid codeword.

The following code illustrates this phenomenon for a single-error-correcting BCH decoder given input with two errors.

n=63;k=57; msg = gf(randint(1, k, 2, 9973)); code = bchenc(msg, n, k);

% Add 2 errors cnumerr2 = zeros(nchoosek(n,2),1); nErrs = zeros(nchoosek(n,2),1); cnumerrIdx = 1; for idx1 = 1 : n-1

sprintf('idx1 for 2 errors = %d', idx1) for idx2 = idx1+1 : n

errors = zeros(1,n); errors(idx1) = 1; errors(idx2) = 1; erroredCode = code + gf(errors); [decoded2, cnumerr2(cnumerrIdx)]...

= bchdec(erroredCode, n, k);

2-21

bchdec

% If bchdec thinks it corrected only one error, % then encode the decoded message. Check that % the re-encoded message differs from the errored % message in only one coordinate. if cnumerr2(cnumerrIdx) == 1

code2 = bchenc(decoded2, n, k); nErrs(cnumerrIdx) = biterr(double(erroredCode.x),...

double(code2.x));

end

cnumerrIdx = cnumerrIdx + 1;

end

% Plot the computed number of errors, based on the difference % between the double-errored codeword and the codeword that was % re-encoded from the initial decoding. plot(nErrs) title(['Number of Actual Errors between Errored Codeword and' ...

'Re-encoded Codeword'])

2-22

The resulting plot shows that all inputs with two errors are decoded to a codeword that differs in exactly one position.

bchdec

Examples The script below encodes a (random) message, simulates the addition of

noise to the code, and then decodes the message.

m=4;n=2 k=5;%M nwords msg = gf % Find t

oly,t] = bchgenpoly(n,k);

[genp % Defi t2 = t;

%Enc code %Cor noi

ne t2, the number of errors to add in this example.

ode the message. = bchenc(msg,n,k); rupt up to t2 bits in each codeword.

sycode = code + randerr(nwords,n,1:t2);

^m-1; % Codeword length

essage length

= 10; % Number of words to encode

(randint(nwords,k)); , the error-correction capability.

2-23

bchdec

% Decode the noisy code. [newmsg,err,ccode] = bchdec(noisycode,n,k); if ccode==code

disp('All errors were corrected.') end if newmsg==msg

disp('The message was recovered perfectly.') end

In this case, all errors are corrected and the message is recovered perfectly. However, if you change the definition of

t2 = t+1;

then some codewords w ill contain more than t errors. This is too many errors, and some are not corrected.

Algorithm bchdec uses the Berlekamp-Massey decoding algorithm. For

information about this algorithm, see the works listed in “References” on page 2-24.

t2 to

Limitations The maximum allowable value of n is 65535.

See Also bchenc, bchgenpoly,“BlockCoding”

References [1] Wicker, Stephen B., Error Control Systems for Digital

Communication and Storage,UpperSaddleRiver,NJ,PrenticeHall,

1995.

[2] Berlekamp, Elwyn R., Algebraic Coding Theory,NewYork, McGraw-Hill, 1968.

2-24

bchenc

Purpose BCH encoder

Note bchenc will be removed in a future release. Use fec.bchenc

instead.

Syntax code = bchenc(msg,n,k)

code = bchenc(...,paritypos)

Description code = bchenc(msg,n,k) encodes the mess ag e in msg using an [n,k]

BCH encoder with the narrow-sense generator polynomial. a Galois array o f symbols over GF(2). Each represents a mes sage word, where the leftmost symbol is the most significant symbol. Parity symbols are at the end of each word in the output Galois array

code = bchenc(...,paritypos) specifies whether bchenc appends or

prepends the parity symbols to the input message to form string

'end'.

paritypos can be either 'end' or 'beginning'. The default is

code.

k-element row of msg

msg is

code.The

The tables below list valid [ the corresponding values of the error-correction capability,

15 7 2

15 5 3

n,k] pairs for small values of n,aswellas

11 1

2-25

bchenc

31 26

31 21 2

31 16 3

31 6 7

63 57

63 51 2

63 45 3

63 39

63 36 5

63 30 6

63 24 7

63 18 10

63 16

63 10 13

63715

2-26

bchenc

127 120

127 113 2

127 106 3

127 99

127 92 5

127 85 6

127 78 7

127 71 9

127 64 10

127 57

127 50 13

127 43

127 36 15

127 29 21

127 22 23

127 15 27

127 8 31

255 247

255 239 2

255 231 3

255 223

2-27

bchenc

255 215 5

255 207 6

255 199 7

255 191 8

255 187 9

255 179 10

255 171

255 163 12

255 155 13

255 147

255 139 15

255 131 18

255 123 19

255 115 21

255 107 22

255 99 23

255 91 25

255 87 26

255 79 27

255 71 29

255 63 30

255 55 31

255 47 42

255 45 43

255 37 45

2-28

bchenc

255 29 47

255 21 55

255 13 59

255 9 63

511 502

511 493 2

511 484 3

511 475

511 466 5

511 457 6

511 448 7

511 439 8

511 430 9

511 421 10

511 412

511 403 12

511 394 13

511 385

511 376 15

511 367 16

511 358 18

2-29

bchenc

511 349 19

511 340 20

511 331 21

511 322 22

511 313 23

511 304 25

511 295 26

511 286 27

511 277 28

511 268 29

511 259 30

511 250 31

511 241 36

511 238 37

511 229 38

511 220 39

511 211

511 202 42

511 193 43

511 184 45

511 175 46

511 166 47

511 157 51

511 148 53

511 139 54

2-30

bchenc

511 130 55

511 121 58

511 112 59

511 103 61

511 94 62

511 85 63

511 76 85

511 67 87

511 58 91

511 49 93

511 40 95

51131109

511 28

51119119

51110121

111

Examples Seetheexampleonthereferencepageforthefunctionbchdec.

Limitations The maximum allowable value of n is 65535.

See Also bchdec, bchgenpoly, bchnumerr,“BlockCoding”

2-31

bchgenpoly

Purpose Generator polynomial of BCH code

Syntax genpoly = bchgenpoly(n,k)

genpoly = bchgenpoly(n,k,prim_poly) [genpoly,t] = bchgenpoly(...)

Description genpoly = bchgenpoly(n,k) returns the narrow-sense generator

polynomial of a BCH code with codeword length

k. The codeword length n must have the form 2

The output

genpoly is a Galois row vector in GF(2) that represents the

coefficients o f the generator polynomial in order of descending powers. The narrow-sense generator polynomial is LCM[m_1(x), m_2(x), ..., m_2t(x)], where LCM is the least common multiple, m_i(x) is the minimum polynomial corresponding to α primitive polynomial for the field GF(

, α is a root of the default

n+1), and t is the error-correcting

capability of the code.

Note Although the bchgenpoly function performs intermediate computations in GF( The output from in GF(

n+1).

n+1), the final polynomial has binary coefficients.

bchgenpoly is a Galois vector in GF(2) rather than

n and message length

-1 for some integer m.

genpoly = bchgenpoly(n,k,prim_poly) is the same as the syntax

above, except that GF(

n+1) that has A as a root. prim_poly is an integer whose binary

prim_poly specifies the pr imitive polynomial for

representation indicates the co efficients of the primitive polynomial. To use the default primitive polynomial for GF(

[genpoly,t] = bchgenpoly(...) returns t, the error-correction

n+1), set prim_poly to [].

capability of the code.

Examples The results below show that a [15,11] BCH code can correct one error

and has a generator polynomial X

m=4;

2-32

+X+1.

n = 2^m-1; % Codeword length k = 11; % Message length % Get generator polynomial and error-correction capability. [genpoly,t] = bchgenpoly(n,k)

The output is

genpoly = GF(2) array.

Array elements =

10011

Limitations The maximum allowable value of n is 511.

bchgenpoly

See Also bchenc, bchdec, bchnumerr,“BlockCoding”

References [1] Peterson, W. Wesley, and E. J. Weldon, Jr., Error-Correcting Codes,

2nd ed., Cambridge, MA, MIT Press, 1972.

2-33

bchnumerr

Purpose Number of correctable errors for BCH code

Syntax T = bchnumerr(N)

T = bchnumerr(N, K)

Description T = bchnumerr(N) returns all the possible combinations of message

length, codeword length, between 3 and 16. lists

T = bchnumerr(N, K) returns the number of correctable errors, t,

for an

See Also bchenc, bchdec, bchgenpoly

K, and number of correctable errors, t, for a BCH code of

N. N must have the form 2

T is a matrix with three columns. The first column

N, the second column lists K, and the third column lists t.

(N, K) BCH code.

-1 for some integer, m ,

2-34

Purpose Bit error rate (BER) for uncoded AWGN channels

Syntax ber = berawgn(EbNo,'pam',M)

ber = berawgn(EbNo,' ber = berawgn(EbNo,' ber = berawgn(EbNo,' ber = berawgn(EbNo,' ber = berawgn(EbNo,' ber = berawgn(EbNo,' ber = berawgn(EbNo,' ber = berawgn(EbNo,' berlb = berawgn(EbNo,' [BER,SER] = berawgn(EbNo, ...)

qam',M) psk',M,dataenc) oqpsk',dataenc) dpsk',M) fsk',M,coherence) fsk',2,coherence,rho) msk',precoding) msk',precoding,coherence)

cpfsk',M,modindex,kmin)

berawgn

Graphical Interface

As an alternative to the berawgn function, invoke the BERTool GUI (

bertool), and u se the Theoretical tab.

Description For All Syntaxes

The berawgn function returns the BER of various modulation schemes over an additive white Gaussian noise (AWGN) channel. The first input argument, density, in dB. If size, whose elements corres po nd to the different E supported modulation schemes, which corre spond to the second input argument to the function, are in the following table.

Modula

Phase

Offs keyi

Dif (DP

tion Scheme

shift keying (PSK)

et quaternary phase shift

ng (OQPSK)

ferential phase shift keying

SK)

EbNo, is the ratio of bit energy to noise power spectral

EbNo is a vector, the output ber is a vector of the same

Second

'psk'

'oqp

'dp

levels. The

b/N0

Input Argument

2-35

berawgn

Modulation Scheme Second Input Argument

Pulse amplitude modulation

'pam'

(PAM)

Quadrature amplitude

'qam'

modulation (QAM)

Frequency shift keying (FSK)

Minimum shift keying (MSK)

Continuous phase frequency shift

'fsk'

'msk'

'cpfsk'

keying (CPFSK)

Most syntaxes also have an M inputthatspecifiesthealphabetsize for the modulation.

k. For all cases, the function assumes the use of a Gray-coded signal

M must have the form 2

for some positive integer

constellation.

For Specific Syntaxes

ber = berawgn(EbNo,'pam',M) returns the BER of uncoded PAM over

an AWGN channel with coherent demodulation.

ber = berawgn(EbNo,'qam',M) returns the BER of uncoded QAM over

an AWGN channel with coherent demodulation. The alphabet size,

must be at least 4. When

of size

ber = berawgn(EbNo,'psk',M,dataenc) returns the BER of coherently

MIJ=×

is used, where

kM= log

detected uncoded PSK over an AWGN channel.

'diff' for differential data encoding or 'nondiff' for nondifferential

data encoding. If

ber = berawgn(EbNo,'oqpsk',dataenc) returns the B ER of coherently

dataenc is 'diff', M must be no greater than 4.

is odd, a rectangular constellation

Ik=−2

and

Jk=+2

dataenc is either

detected offset-QPSK over an uncoded AWGN channel.

ber = berawgn(EbNo,'dpsk',M) returns the BER of uncoded DPSK

modulation over an AWGN channel.

2-36

berawgn

ber = berawgn(EbNo,'fsk',M,coherence) returns the BER of

orthogonal uncoded FSK modulation over an AWGN channel.

coherence is either 'coherent' for coherent demodulation or 'noncoherent' for noncoherent demodulation. M must be no greater

than 64 for

ber = berawgn(EbNo,'fsk',2,coherence,rho) returns the BER for

binary nonorthogonal FSK over an uncoded AWGN channel, where

rho is the complex correlation coefficient. See “Nonorthogonal 2-FSK

with Coherent Detection” for the definition of the complex correlation coefficient and how to compute it for nonorthogonal BFSK .

ber = berawgn(EbNo,'msk',precoding) returns the BER of coherently

detected MSK modulation over an uncoded AWGN channel. Setting

precoding to 'off' returns results for conventional MSK while setting precoding to 'on' returns results for precoded MSK.

ber = berawgn(EbNo,'msk',precoding,coherence) specifies whether

the detection is coherent or noncoherent.

berlb = berawgn(EbNo,'cpfsk',M,modindex,kmin) returns a low er

bound on the BER of uncoded CPFSK modulation over an AWGN channel.

kmin is the number of paths having the minimum distance; if this

number is unknown, you can assume a value of 1.

'noncoherent'.

modindex is the modulation index, a positive real number.

[BER,SER] = berawgn(EbNo, ...) returns both the BER and SER.

Examples An example using this function is in “Comparing Theoretical and

Empirical Error Rates”.

Limitations The numerical accuracy of this function’s output is limited by

approximations related to the numerical implementation of the expressions.

You can generally rely on the first couple of significant digits of the function’s output.

2-37

berawgn

See Also bercoding, berfading, bersync, “Theoretical Performance Results”,

Analytical Expressions U sed in berawgn, bercoding, berfading, and BERTool

References [1] Anderson, John B., Tor Aulin, and Carl-Erik Sundberg, Digital

Phase Modulation,NewYork,PlenumPress,1986.

[2] Cho, K., and Yoon, D., “On the general BER expression of one- and two-dimensional amplitude modulations”, IEEE Trans. Commun.,Vol. 50, Number 7, pp. 1074-1080, 2002.

[3] Lee, P. J., “Computation of the bit error rate of coherent M-ary PSK with Gray code bit mapping”, IEEE Trans. Commun., Vol. COM-34, Number 5, pp. 488-491, 1986.

[4] Proakis,J.G.,Digital Communications, 4th ed., McGraw-Hill, 2001.

[5] Simon, M. K, Hinedi, S. M ., and Lindsey, W . C., Digital Communication Techniques – Signal Design and Detection, Prentice-Hall, 1995.

2-38

[6] Simon, M. K, “On the bit-error probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization”, IEEE Trans. Commun., Vol. 54, pp. 806-812, 2006.

[7] Lindsey, W. C., and Simon, M. K, Telecommunication Systems Engineering, Englewood Cliffs, N.J., Prentice-Hall, 1973.

bercoding

Purpose Bit error rate (BER) for coded AWGN channels

Syntax berub = bercoding(EbNo,'conv',decision,coderate,dspec)

berub = bercoding(EbNo,' berub = bercoding(EbNo,' berapprox = bercoding(EbNo,' berub = bercoding(EbNo,' berapprox = bercoding(EbNo,'

block','hard',n,k,dmin) block','soft',n,k,dmin)

Hamming','hard',n)

Golay','hard',24)

RS','hard',n,k)

Graphical Interface

As an alternative to the bercoding function, invoke the BERTool GUI (

bertool) and use the Theoretical tab.

Description berub = bercoding(EbNo,'conv',decision,coderate,dspec)

returnsanupperboundorapproximationontheBERofabinary convolutional code with coherent phase shift keying (PSK ) modulation over an additive white Gaussian noise (AWGN) channel. ratio of bit energy to noise power spectral density, in dB. If vector, to the different E

decision to 'hard'; to specify soft-decision decoding, set decision to 'soft'. The convolutional code has code rate equal to coderate.The dspec input is a structure that contains information about the code’s

berub is a vector of the same size, whose elem ents correspond

levels. To specify hard-decision decoding, set

b/N0

distance spectrum:

•

dspec.dfree is the minimum free distance of the code.

dspec.weight is the weight spectrum of the code.

•

To find distance spectra for some sample codes, use the function or see [5] and [3].

Note The results for binary PSK and quaternary PSK modula tion are the same. This function does not support M-ary PSK when M is other than 2 or 4.

EbNo is the

EbNo is a

distspec

2-39

bercoding

berub = bercoding(EbNo,'block','hard',n,k,dmin) returns an

upper bound on the BER of an [ decoding and coherent BPSK or Q PSK modulation.

n,k] binary block code with hard-decision

dmin is the

minimum distance of the code.

berub = bercoding(EbNo,'block','soft',n,k,dmin) returns an

upper bound on the BER of an [ decoding and coherent BPSK or Q PSK modulation.

n,k] binary block code with soft-decision

dmin is the

minimum distance of the code.

berapprox = bercoding(EbNo,'Hamming','hard',n) returns an

approximation of the BER of a Hamming code using hard-decision decoding and coherent BPSK modulation. (For a Hamming code, if n is known, then k can be computed directly from n.)

berub = bercoding(EbNo,'Golay','hard',24) returns an upper

bound of the BER of a Golay code using hard-decision decoding and coherent BPSK modulation. Support for Golay currently is only for n=24. In accordance with [3], the Golay coding upper bound assumes only the correction of 3-error patterns. Even though it is theoretically possible to correct approximately 19% of 4-error patterns, most decoders in practice do not have this capability.

berapprox = bercoding(EbNo,'RS','hard',n,k) returns an

approximation of the BER of (n,k) Reed-Solomon code using hard-decision decoding and coherent BPSK modulation.

Examples An example using this function for a convolutional code is in “Plotting

Theoretical Error Rates”.

The following example finds an upper bound on the theoretical BER of a block code. It also uses the

n = 23; k = 12; % Lengths of codewords and messages dmin = 7; % Minimum distance EbNo = 1:10; ber_block = bercoding(EbNo,'block','hard',n,k,dmin); berfit(EbNo,ber_block) % Plot BER points and fitted curve. ylabel('Bit Error Probability');

2-40

berfit function to perform curve fitting.

bercoding

title('BER Upper Bound vs. Eb/No, with Best Curve Fit');

Limitations The numerical accuracy of this function’s output is limited by

• Approximati on s in the analysis l ea ding to the closed-form expressions

that the function uses

• Approximations related to the numerical implementation of the

expressions

You can generally rely on the first couple of significant digits of the function’s output.

See Also berawgn, berfading, bersync, distspec, “Theoretical Performance

Results” Analytical Expressions Usedinberawgn,bercoding,berfading, and BERTool

2-41

bercoding

References [1] Proakis, J. G., Digital Communications,4thed.,NewYork,

McGraw-Hill, 2001.

[2] Frenger, P., P. Orten, and T. Ottosson, “Convolutional Codes with Optimum Distance Spectrum,” IEEE Communications Letters,Vol. 3, No. 11, Nov. 1999, pp. 317–319.

[3] Odenwalder, J. P., Error Control Coding Handbook,FinalReport, LINKABIT Corporation, San Diego, CA, 1976.

[4] Sklar, B., Digital Communications, 2nd ed., Prentice Hall, 2001.

[5] Ziemer, R. E., and R. L., Peterson, Introduction to Digital Communication, 2nd ed., Prentice Hall, 2001.

2-42

berconfint

Purpose Bit error rate (BER) and confidence interval of Monte Carlo simulation

Syntax [ber,interval] = berconfint(nerrs,ntrials)

[ber,interval] = berconfint(nerrs,ntrials,level)

Description [ber,interval] = berconfint(nerrs,ntrials) returns the error

probability estimate Monte Carlo simulation of is a two-element vector that lists the endpoints of the interval. If the errors and trials are measured in bits, the error probability is the bit error rate (BER); if the errors and trials are measured in symbols, the error probability is the symbol error rate (SER).

[ber,interval] = berconfint(nerrs,ntrials,level) specifies the

confidence level as a real number between 0 and 1.

ber and the 95% confidence interval interval for a

ntrials trials with nerrs errors. interval

Examples If a simulation of a communication system results in 100 bit errors in

trials, the BER (bit error rate) for that simulation is the quotient

-4

. The command below finds the 95% confidence interval for the

BER of the system.

nerrs = 100; % Number of bit errors in simulation ntrials = 10^6; % Number of trials in simulation level = 0.95; % Confidence level [ber,interval] = berconfint(nerrs,ntrials,level)

The output below shows that, with 95% confidence, the BER for the system is between 0.0000814 and 0.0001216.

ber =

1.0000e-004

interval =

1.0e-003 *

2-43

berconfint

0.0814 0.1216

For an example that uses the output of berconfint to plot error bars on a BER plot, see “Example: Curve Fitting for an Error Rate Plot”

See Also binofit (Statistics Toolbox), mle (Statistics Toolbox), “Performance

Evaluation”

References [1] Jeruchim, Michel C., Philip Balaban, and K. Sam Shanmugan,

Simulation of Communication Systems, Second Edition, New York, Kluwer Academic/Plenum, 2000.

2-44

Purpose Bit error rate (BER) for Rayleigh and Rician fading channels

Syntax ber = berfading(EbNo,'pam',M,divorder)

ber = berfading(EbNo,'qam',M,divorder) ber = berfading(EbNo,'psk',M,divorder) ber = berfading(EbNo,'depsk',M,divorder) ber = berfading(EbNo,'oqpsk',divorder) ber = berfading(EbNo,'dpsk',M,divorder) ber = berfading(EbNo,'fsk',M,divorder,coherence) ber = berfading(EbNo,'fsk',2,divorder,coherence,rho) ber = berfading(EbNo,...,K) ber = berfading(EbNo,'psk',2,1,K,phaserr) [BER,SER] = berfading(EbNo, ...)

berfading

Graphical Interface

As an alternative to the berfading function, invoke the BERTool GUI (

bertool), and u se the Theoretical tab.

Description For All Syntaxes

The first input argument, EbNo, is the ratio of bit energy to noise power spectral density, in dB. If thesamesize,whoseelementscorrespondtothedifferentE

Most syntaxes also have a n the modulation.

berfading uses expressions that assume Gray coding. If you use binary

coding, the results may differ.

For cases where diversity is used, the SNR on each diversity branch is

EbNo/divorder,wheredivorder is the diversity order (the number of

diversity branches) and is a positive integer.

For Specific Syntaxes

ber = berfading(EbNo,'pam',M,divorder) returns the BER for PAM

over an uncoded Rayleigh fading channel with coherent demodulation.

ber = berfading(EbNo,'qam',M,divorder) returns the BER for QAM

over an uncoded Rayleigh fading channel with coherent dem odulation.

M must have the form 2

EbNo is a vector, the output ber is a vector of

levels.

b/N0

M inputthatspecifiesthealphabetsizefor

for some positive integer k.

2-45

berfading

The alphabet size, M,mustbeatleast4. When

rectangular constellation of size

Jk=+2

and

ber = berfading(EbNo,'psk',M,divorder) returns the BER for

MIJ=×

kM= log

is used, where

is odd, a

Ik=−2

coherently detected PSK over an uncoded Rayleigh fading channel.

ber = berfading(EbNo,'depsk',M,divorder) returns the BER

for coherently detected PSK with differential data encoding over an uncoded Rayleigh fading channel. Only M = 2 is currently supported.

ber = berfading(EbNo,'oqpsk',divorder) returns the BER of

coherently detected offset-QPSK over an uncoded Rayleigh fading channel.

ber = berfading(EbNo,'dpsk',M,divorder) returns the BER for

DPSK over an uncoded Rayleigh fading channel. For DPSK, it is assumed that the fading is slow enough that two consecutive symbols are affected by the same fading coefficient.

ber = berfading(EbNo,'fsk',M,divorder,coherence) returns

the BER for orthogonal FSK over an uncoded Rayleigh fading channel.

'noncoherent' for noncoherent detection.

coherence should be 'coherent' for coherent detection, o r

2-46

ber = berfading(EbNo,'fsk',2,divorder,coherence,rho) returns

the BER for binary nonorthogonal FSK over an uncoded Rayleigh fading channel.

rho is the complex correlation coefficient. See “Nonorthogonal

2-FSK with Coherent Detection” for the definition of the complex correlation coefficient and how to compute it for nonorthogonal BFSK.

ber = berfading(EbNo,...,K) returns the BER over an uncoded

Rician fading channel, where in linear scale. For the case of

ber = berfading(EbNo,'psk',2,1,K,phaserr) returns the BER of

K is the ratio of specular to diffuse energy

'fsk', rho must be specified before K.

BPSK over an uncoded Rician fading channel with imperfect phase

berfading

synchronization. phaserr is the standard deviation of the reference carrier phase error in radians.

[BER,SER] = berfading(EbNo, ...) returns both the BER and SER.

Examples The following example computes and plots the BER for uncoded DQPS K

(differential quaternary phase shift keying) modulation over an flat Rayleigh fading channel.

EbNo = 8:2:20; M = 16; % Use 16 QAM L = 1; % Start without diversity ber = berfading(EbNo,'qam',M,L); semilogy(EbNo,ber); text(18.5, 0.02, sprintf('L=%d', L)) hold on % Loop over diversity order, L, 2 to 20 for L=2:20

ber = berfading(EbNo,'qam',M,L);

semilogy(EbNo,ber); end text(18.5, 1e-11, sprintf('L=%d', L)) title('QAM over fading channel with diversity order 1 to 20') xlabel('E_b/N_o (dB)') ylabel('BER') grid on

2-47

berfading

Limitations The numerical accuracy of this function’s output is limited by

approximations related to the numerical implementation of the expressions

You can generally rely on the first couple of significant digits of the function’s output.

See Also berawgn, bercoding, bersync, “Theoretical Performance Results”

Analytical Expressions U sed in berawgn, bercoding, berfading, and BERTool

roakis, John G., Digital Communications, 4th ed., New York,

[1] P

aw-Hill, 2001.

McGr

odestino, James W., and Mui, Shou Y., Convolutional code

[2] M

formance in the Rician fading channel, IEEE Trans. Commun., 1976.

per

2-48

Refe

rences

berfading

[3] Cho, K., and Yoon, D., “On the general BER expression of one- and two-dimensional amplitude modulations”, IEEE Trans. Commun.,Vol. 50, Number 7, pp. 1074-1080, 2002.

[4] Lee, P. J., “Computation of the bit error rate of coherent M-ary PSK with Gray code bit mapping”, IEEE Trans. Commun., Vol. COM-34, Number 5, pp. 488-491, 1986.

[5] Lindsey, W. C., “Error probabilities for Rician fading multichannel reception of binary and N-ary signals”, IEEE Trans. Inform. Theory, Vol. IT-10, pp. 339-350, 1964.

[6]Simon,M.K,Hinedi,S.M.,andLindsey,W.C.,Digital Communication Techniques – Signal Design and Detection, Prentice-Hall, 1995.

[7] Simon, M. K., and Alouini, M. S., Digital Communication over Fading Channels – A Unified Approach to Performance Analysis,1st ed., Wiley, 2000.

[8] Simon,M.K,“Onthebit-errorprobability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization”, IEEE Trans. Commun., Vol. 54, pp. 806-812, 2006.

2-49

berfit

Purpose Fit curve to nonsmooth empirical bit error rate (BER) data

Syntax fitber = berfit(empEbNo,empber)

fitber = berfit(empEbNo,empber,fitEbNo) fitber = berfit(empEbNo,empber,fitEbNo,options) fitber = berfit(empEbNo,empber,fitEbNo,options,fittype) [fitber,fitprops] = berfit(...) berfit(...) berfit(empEbNo,empber,fitEbNo,options,

Description fitber = berfit(empEbNo,empber) fits a curve to the empirical BER

data in the vector (BER) points. The values in E

values, in dB, given by empEbNo.ThevectorempEbNo must be in

b/N0

ascending order and must have at least four elements.

Note The berfit function is intended for curve fitting or interpolation, not extrapolation. Extrapolating BER data beyond an order of magnitude below the smallest empirical BER value is inherently unreliable.

empber and returns a vector of fitted bit error rate

empber and fitber correspond to the

'all')

2-50

fitber = berfit(empEbNo,empber,fitEbNo) fits a curve to the

empirical BER data in the vector

empber corresponding to the E

b/N0

values, in dB, given by empEbNo. The function then evaluates the curve at the E points. The length of

fitber = berfit(empEbNo,empber,fitEbNo,options) uses the

structure These options are the ones used by the create the

values, in dB, given by fitEbNo and returns the fitted BER

b/N0

options to override the default options used for optimization.

options structure using the optimset function. Particularly

fitEbNo must equal or exceed that of empEbNo.

fminsearch function. You can

relevant fields are described in the table below.

berfit

Field

options.Display

options.MaxFunEvals

options.MaxIter

options.TolFun

options.TolX

Description

Level of display: 'off' (default) displays no output;

'iter'

displays output at each iteration;

'final' displays only the final

output;

'notify' displays output

only if the function does not converge.

Maximum number of function evaluations before optimization ceases. The default is 10

Maximum number of iterations before optimization ceases. The default is 10

Termination tolerance on the closed-form function used to generate the fit. The default is

-4

Termination tolerance on the coefficient values of the closed-form function used to generate the fit. The default is

-4

fitber = berfit(empEbNo,empber,fitEbNo,options,fittype)

specifies which closed-form function berfit uses to fit the empirical data, from the possible fits listed in “Algorithm” on page 2-53 below.

'doubleExp+const'. To avoid overriding default optimization options,

use

[fitber,fitprops] = berfit(...) returns the MATLAB structure fitprops, which describes the results of the curve fit. Its fields are

fittype can be 'exp', 'exp+const', 'polyRatio',or

options = [].

described in the table below.

2-51

berfit

Field

fitprops.fitType

fitprops.coeffs

fitprops.sumSqErr

fitprops.exitState

fitprops.funcCount

fitprops.iterations

Description

The closed-form function type used to generate the fit:

'exp+const', 'polyRatio',or 'doubleExp+const'.

'exp',

Thecoefficientsusedtogenerate the fit. If the function cannot find a valid fit,

fitprops.coeffs

is an empty vector.

The sum squared error between the log of the fitted BER points and the log of the empirical BER points.

The exit condition of berfit:

'The curve fit converged to a solution.' maximum number of function evaluations was exceeded.'

'No desirable fit was

found'

, 'The

The number of function evaluations used in minimizing the sum squared error function.

The number of iterations taken in minimizing the sum squared error function. This is not necessarily equal to the numb er of function evaluations.

2-52

berfit(...) plots the empirical and fitted BER data.

berfit(empEbNo,empber,fitEbNo,options,'all') plots the empirical

and fitted BER data from all the possible fits, listed in the “Algorithm”

berfit

on page 2-53 below, that return a valid fit. To avoid overriding default options, use

Note Avalidfitmustbe

• real-valued

• monotonically decreasing

• greater than or equal to 0 and less than or equal to 0.5

If a fit does not confirm to this criteria, it is rejected.

Algorithm The berfit function fits the BER data using unconstrained nonlinear

optimization via the that

berfit considers are listed in the table below, where x is the

b/N0

functions were empirically found to provide close fits in a wide variety of situations, including exponentiallydecayingBERs,linearlyvarying BERs, and BER curves with error rate floors.

options = [].

fminsearch function. The closed-form functions

in linear terms (not dB) and f is the estimated BER. These

Value of f ittype

'exp'

'exp+const'

Functional Expression

()

axa

exp{ [( ) ]}

−−

axa

exp[ ( ) ]

−−

2-53

berfit

Value of f ittype

'polyRatio'

'doubleExp+const'

Functional Expression

ax ax a

()=

xaxaxa

⎡

axa

exp

−−

()

⎢

⎣

axa

exp

+++

⎤ ⎥

⎦

⎡

−−

()

⎢

⎣

⎤ ⎥

⎦

The sum squared error function that fminsearch attempts to minimize is

F =−

[log( ) log( )]empirical BER fitted BER

∑

where the fitted BER points are the values in fitber and the sum is over the E

points given in empEbNo. It is important to use the

b/N0

log of the BER values rather than the BER values themselves so that the high-BER regions do not dominate the objective function inappropriately.

Examples The examples below illustrate the syntax of the function, but they use

hard-coded or theo retical BER data for simplicity. For an example that uses empirical BER data from a simulation, see “Example: Curve Fitting for an Error Rate Plot”.

The code below plots the best fit for a sample set of data.

EbNo = 0:13;

berdata = [.2 .15 .13 .12 .08 .09 .08 .07 .06 .04 .03 .02 .01 .004];

berfit(EbNo,berdata); % Plot the best fit.

2-54

berfit

The curve at the va like uses mor fit exp

The nex floor. with a MMSE e optio

'exp 'pol

EbNo = -10:3:15;

empBER = [0.3361 0.3076 0.2470 0.1878 0.1212 0.0845 0.0650 0.0540 0.0474];

connects the points created by evaluating the fit expression

lues in

fit(EbNo,berdata,[0:0.2:13])

ber

EbNo. To make the curve look smoother, use a syntax

. This alternative syntax

e points when plotting the curve, but it does not change the

ression.

t example demonstrates a fit for a BER curve with an error

We generate the empirical BER array by simulating a channel

null (

ch = [0.5 0.47]) with BPSK modulation and linear

qualizer at the receiver. We run the berfit with the

n. The

' yRatio'

'doubleExp+const' fit does not provide a valid fit, and the

fit type does not work well for this data. The 'exp+const' and

fits closely match the simulated data.

'all'

2-55

berfit

figure; berfit(EbNo, empBER, [], [], 'all');

2-56

The following code illustrates the use of the options input structure as well as the

fitprops output structure. The 'notify' value for the

display level causes the function to produce output when one of the attempted fits does not converge. The

exitState field of the output

structure also indicates which fit converges and which fit does not.

M = 4; EbNo = 3:10; berdata = berfading(EbNo,'psk',M,2); % Compute theoretical BER. noisydata = berdata.*[.93 .92 1 .59 .08 .15 .01 .01]; % Say when fit fails to converge. options = optimset('display','notify');

disp('*** Trying exponential fit.') % Poor fit [fitber1,fitprops1] = berfit(EbNo,noisydata,EbNo,...

berfit

options,'exp')

disp('*** Trying polynomial ratio fit.') % Good fit [fitber2,fitprops2] = berfit(EbNo,noisydata,EbNo,...

options,'polyRatio')

Theoutputisasfollows:

*** Trying exponential fit.

Exiting: Maximum number of function evaluations has been exceeded

- increase MaxFunEvals option. Current function value: 2.729948

fitber1 =

0.0766 0.0423 0.0205 0.0086 0.0030 0.0009 0.0002 0.00

fitprops1 =

fitType: 'exp'

coeffs: [4x1 double]

sumSqErr: 2.7299

exitState: 'The maximum number of function evaluations...

has been exceeded'

funcCount: 10000

iterations: 6177

*** Trying polynomial ratio fit.

fitber2 =

0.0931 0.0476 0.0220 0.0090 0.0031 0.0008 0.0001 0.00

2-57

berfit

fitprops2 =

fitType: 'polyRatio'

coeffs: [6x1 double]

sumSqErr: 2.0578 exitState: 'The curve fit converged to a solution' funcCount: 580

iterations: 344

See Also fminsearch, optimset, “Performance Evaluation”

References For a general description of unconstrained nonlinear optimization, see

the following work.

[1] Chapra, Steven C., and Raymond P. Canale, Numerical Methods for Engineers, Fourth Edition, New York, McGraw-Hill, 2002.

2-58

Purpose Bit error rate (BER) for imperfect synchronization

Syntax ber = bersync(EbNo,timerr,'timing')

ber = bersync(EbNo,phaserr,

'carrier')

bersync

Graphical Interface

As an alternative to the bersync function, invoke the BERTool GUI (

bertool) and use the Theoretical tab.

Description ber = bersync(EbNo,timerr,'timing') returns the BER of uncoded

coherent binary phase shift keying (BPSK) m odulation over an additive white Gaussian noise (AWGN) channel with imperfect timing. The normalized timing error is assumed to have a Gaussian distribution.

EbNo is the ratio of bit energy to noise power spectral density, in

dB. If

EbNo is a vector, the output ber is a vector of the same size,

whose elements correspond to the different E the standard deviation of the timing error, normalized to the symbol interval.

ber = bersync(EbNo,phaserr,'carrier') returns the BER of uncoded

BPSK modulation over an AWGN channel with a noisy phase re fere nce. The phase error is assumed to have a Gaussian distribution. is the standard deviation of the error in the reference carrier phase, in radians.

timerr must be between 0 and 0.5.

levels. timerr is

b/N0

phaserr

Examples The code below computes the BER of coherent BPSK modulation over an

AWGN channel with imperfect timing. The example varies both and timerr.(Whentimerr assumes the final value of zero, the bersync command produces the same result as berawgn(EbNo,'psk',2).)

EbNo = [4 8 12]; timerr = [0.2 0.07 0]; ber = zeros(length(timerr), length(EbNo)); for ii = 1:length(timerr)

ber(ii,:) = bersync(EbNo, timerr(ii),'timerr'); end % Display result using scientific notation. format short e; ber

EbNo

2-59

bersync

format; % Switch back to default notation format.

The output is below, where each row c orresponds to a different value of

timerr and each column corresponds to a different value of EbNo.

ber =

5.2073e-002 2.0536e-002 1.1160e-002

1.8948e-002 7.9757e-004 4.9008e-006

1.2501e-002 1.9091e-004 9.0060e-009

Limitations The numerical accuracy of this function’s output is limited by

• Approximati on s in the analysis l ea ding to the closed-form expressions

that the function uses

• Approximations related to the numerical implementation of the

expressions

You can generally rely on the first couple of significant digits of the function’s output.

Limitations Related to Extreme Values of Input Arguments

Inherent limitations in numerical precision force the function to assume perfect synchronization if the value of small. The table below indicates how the function behaves under these conditions.

Condition

timerr < eps bersync(EbNo,timerr,'timing')

phaserr < eps bersync(EbNo,phaserr,'carrier')

Algorithm This function uses formulas from [3].

When the last input is

2-60

'timing', the function computes

timerr or phaserr is very

Behavior of Function

defined as berawgn(EbNo,'psk',2)

bersync

∞

exp( ) exp( ) exp( )

∫∫

−∞

πσ

∞

−−+−

R 22R

212

()

−

dxd

∞

∫

where σ is the timerr input and R is the value of EbNo converted from dB to a linear scale.

When the last input is

∞∞

exp( ) exp( )

∫∫

πσ

−−

'carrier', the function computes

cos

dyd

where σ is the phaserr input and R is the value of EbNo converted from dB to a linear scale.

See Also berawgn, bercoding, berfading, “Theoretical Performance Results”

References [1] Jeruchim, Michel C., Philip Balaban, and K. Sam Shanmugan,

Simulation of Communication Systems, Second Edition, New York, Kluwer Academic/Plenum, 2000.

[2] Sklar, Bernard, Digital Communications: Fundamentals and Applications, Second Edition, Upper Saddle River, NJ, Prentice-Hall,

2001.

[3] Stiffler, J. J., Theory of Synchronous Communications,Englewood Cliffs, NJ, Prentice-Hall, 1971.

2-61

bertool

Purpose Open bit error rate analysis GUI (BERTool)

Syntax bertool

Description bertool launches the Bit Error Rate Analysis Tool (BERTool). BERTool

is a graphical user interface (GUI) that enables you to analyze BER performance of communications systems. Performance analysis is done via simulation-based, semianalytic, or theoretical approach. To learn more abo ut BERTool, s ee “BERTool: A Bit Error Rate Analysis GUI”.

2-62

bi2de

Purpose Convert binary vectors to decimal numbers

Syntax d = bi2de(b)

d = bi2de(b,flg) d = bi2de(b,p) d = bi2de(b,p,flg)

Description d = bi2de(b) converts a binary row vector b to a nonnegative decimal

integer. If number. In this case, the output which is the decimal representation of the corresponding row of

Note By default, bi2de interprets the first column of b as the lowest-order digit.

d = bi2de(b,flg) is the same as the syntax above, except that flg

is a string that determines whether the first column of b contains the lowest-order or highest-order digits. Possible values for

'right-msb' and 'left-msb'.Thevalue'right-msb' produces the

default behavior.

b is a matrix, each row is interpreted separately as a binary

d is a column vector, each element of

flg are

d = bi2de(b,p) converts a base-p ro w vector b to a nonnegative

decimal integer , where first column of

d is a nonnegative decimal vector, each row of which is the decimal

b is the lowest base-p digit. If b is a matrix, the output

form of the corresponding row of

d = bi2de(b,p,flg) is the same as the syntax above, except that flg is a string that determines whether the first column of b contains

the lowest-order or highest-order digits. Possible values for

'right-msb' and 'left-msb'.Thevalue'right-msb' produces the

p is an integer greater than or equal to 2. The

flg are

default behavior.

2-63

bi2de

Examples The code below generates a matrix that contains binary representations

of five random numbers between 0 and 15. It then converts all five numbers to decimal integers.

b = randint(5,4); % Generate a 5-by-4 random binary matrix. de = bi2de(b); disp(' Dec Binary') disp(' ----- -------------------') disp([de, b])

Sample output is below. Your results might vary because the numbers are random.

Dec Binary

----- ------------------131011

71110

151111

40010 91001

Comparing a Two-Dimensional Matrix x with Another Input y

biterr

Shape of y flg Type of

Comparison

2-D m a trix

'overall'

(default)

'row-wise'

Element by element

mth row of x vs. mth row of

'column-wise'

mth column of

x vs. mth

column of

number Total

Total number of bit errors

Column vector whose entries count bit errors in each row

Row vector whose

entries count bit errors in each column

Number of Bits

k times

number of entries of

k times

number of entries of

k times

number of entries of

2-69

biterr

Comparing a Two-Dimensional Matrix x with Another Input y(Continued)

Shape of y flg Type of

Comparison

Row vector

Column vector

'overall'

'row-wise'

(default)

'overall'

'column-wise'

(default)

vs. each

row of

y vs. each

row of

vs. each

column of

y vs. each

column of

number Total

Total number of bit errors

Column vector whose entries count bit errors in each row of

Total number

of bit errors

Row vector

whose entries count bit errors in each column of

Number of Bits

k times

number of entries of

times size

k times

number of entries of

times size

2-70

[number,ratio,individual] = biterr(...) returns a matrix individual whose dimensions are those of the larger of x and y.Each

entry of individual corresponds to a comparison between a pair of elements of elements in the pair differ.

Examples Example 1

The commands below compare the column vector [0; 0; 0] to each column of a random binary matrix. The output is the number, proportion, and locationsof1sinthematrix. Inthiscase, the random matrix.

format rat; [number,ratio,individual] = biterr([0;0;0],randi([0 1],3,5))

The output is

number =

20031

biterr

x and y, and specifies the number of bits by w hich the

individual isthesameas

ratio =

2/3 0 0 1 1/3

individual =

10010 10010 00011

Example 2

The commands below illustrate the use of flg to override the default row-by-row comparison.

number and ratio are scalars, and individual

has the same dimensions as the larger of the first two arguments of

biterr.

2-71

biterr

format rat; [number2,ratio2,individual2] = biterr([1 2; 3 4],[1 3],3,'overall')

The output is

number2 =

ratio2 =

5/12

individual2 =

01 13

2-72

Example 3

The script below adds errors to 10% of the elements in a matrix. Each entry in the matrix is a two-bit nu mber in de c imal form. The script computes the bit error rate using using

symerr.

x = randi([0 3],100); % Original signal % Create errors to add to ten percent of the elements of x. % Errors can be either 1, 2, or 3 (not zero). errorplace = (rand(100,100) > .9); % Where to put errors errorvalue = randi(3,100); % Value of the errors errors = errorplace.*errorvalue; y = rem(x+errors,4); % Signal with errors added, mod 4 format short [num_bit,ratio_bit] = biterr(x,y,2) [num_sym,ratio_sym] = symerr(x,y)

biterr and the symbol error rate

biterr

Sample output is below. ratio_sym is close to the target value of 0.10. Your results might vary because the example uses random numbers.

num_bit =

1304

ratio_bit =

0.0652

num_sym =

981

ratio_sym =

0.0981

Example 4

Thefollowingexampleuseslogicalinputarguments.

SNR = 3; frameLen = 100; x = randi([0 1], frameLen, 1); y = awgn(2*x-1, SNR); z=y>0; biterr(x, z)

Example 5

Thefollowingexampleuseslogicalinputarguments.

SNR = 5; frameLen = 100; x = rand(100, 1) > 0.5; y = awgn(2*x-1, SNR);

2-73

biterr

z=y>0; biterr(x, z)

See Also symerr, “Performance Results via S imulation”

2-74

Purpose Model binary symmetric channel

Syntax ndata = bsc(data,p)

ndata = bsc(data,p,s) ndata = bsc(data,p,state) [ndata,err] = bsc(...)

Description ndata = bsc(data,p) passes the binary input signal data through

a binary symmetric channel with error probability introduces a bit error with probability

data independently. data must be an array of binary numbers or a

Galois array in GF(2).

bsc(data,p,s)

random stream. See RandStream for more details.

ndata = bsc(data,p,state) resets the state of the uniform random

number generator

Note This usage is deprecated and may be removed in a future release. Instead of

causes rand to use the random stream s. s is any valid

p must be a scalar between 0 and 1.ndata =

rand to the integer state.

state,uses, as in the previous example.

p, processing each element of

p. The channel

bsc

[ndata,err] = bsc(...) returns an array, err,containingthe

channel errors.

This function uses, by default, the Mersenne Twister algorithm by Nishimura and Matsumoto.

Note Using the state parametercausesthisfunctiontoswitch random generators to use the

See

rand for details on the generator algorithm.

'state' algorithm of the rand function.

2-75

bsc

Examples To introduce bit errors in the bits in a random matrix with probability

0.15, use the

z = randint(100,100); % Random matrix

nz = bsc(z,.15); % Binary symmetric channel

[numerrs, pcterrs] = biterr(z,nz) % Number and percentage of errors

The output below is typical. The percentage of bit errors is not exactly 15% in most trials, but it is close to 15% if the size of the matrix large.

numerrs =

pcterrs =

bsc function:

1509

0.1509

z is

Another example using this function is in “Binary Symmetric Channel”.

See Also rand, awgn, “Binary Symmetric Channel”

2-76

Purpose Construct constant modulus algorithm (CMA) object

Syntax alg = cma(stepsize)

alg = cma(stepsize,leakagefactor)

Description The cma function creates an adaptive algorithm object that you can use

with the You can then use the equalizer object with the equalize a signal. To learn more about the process for equalizing a signal, see “Using Adaptive Equalizer Functions and Objects”.

Note After you use either lineareq or dfe to create a CMA equalizer object, you should initialize the equalizer object’s a nonzero vector. Typically, CMA is used with differential modulation; otherwise, the initial weights a re very important. A typical vector of initial weights has a 1 corresponding to the center tap and 0s elsewhere.

lineareq function or dfe function to create an equalizer object.

equalize function to

Weights property with

cma

alg = cma(stepsize) constructs an adaptive algorithm object based

on the constant modulus algorithm (CMA) with a step size of

alg = cma(stepsize,leakagefactor) sets the leakage factor of

the CMA. corresponds to a conventional w eight update algorithm, while a value of 0 corresponds to a memoryless update algorithm.

leakagefactor must be between 0 and 1. A value of 1

stepsize.

Properties

The table below describes the properties of the CMA adaptive algorithm object. Tolearnhowtovieworchangethevaluesofanadaptive algorithm object, see “Accessing Properties of an Adaptive Algorithm”.

2-77

cma

Property

AlgType

StepSize

Description

Fixed value, 'Constant

Modulus'

CMA step size parameter, a nonnegative real number

LeakageFactor

CMA leakage factor, a real number between 0 and 1

Algorithm Referring to the schematics p resented in “Overview of Adaptive

Equalizer Classes”, def in e w as the vector of all weights w u as the vector of all inputs u

. B ased on the current set of weights, w,

this adaptive algorithm creates the new set of weights given by

(

LeakageFactor)w+(StepSize)u

where the * operator denotes the complex conjugate.

and define

See Also lms, signlms, normlms, varlms, rls, lineareq, dfe, equalize,

“Equalizers”

References [1] Haykin, Simon, Adaptive Filter Theory, Third Ed., Upper Saddle

River, NJ, Prentice-Hall, 1996.

2-78

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Mathworks COMMUNICATIONS TOOLBOX 4 Reference

Specifications and Main Features

Frequently Asked Questions

User Manual

Function Reference

Signal Sources

Source Coding

Error-Control Coding

Interleaving/Deinterleaving

Analog Modulation/Demodulation

Digital Modulation/Demodulation

Pulse Shaping

Filters

Channels

Equalizers

MATLAB Functions and Operators

Galois Fields of Odd Characteristic

Utilities

Functions — Alphabetical List

Comparing a Two-Dimensional Matrix x with Another Input y